In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms. The derivation of a theorem is often interpreted as a proof of the truth of the resulting expression, but different deductive systems can yield other interpretations, depending on the meanings of the derivation rules. The proof of a mathematical theorem is a logical argument demonstrating that the conclusions are a necessary consequence of the hypotheses, in the sense that if the hypotheses are true then the conclusions must also be true, without any further assumptions. The concept of a theorem is therefore fundamentally deductive, in contrast to the notion of a scientific theory, which is empirical.

Although they can be written in a completely symbolic form using, for example, propositional calculus, theorems are often expressed in a natural language such as English. The same is true of proofs, which are often expressed as logically organized and clearly worded informal arguments, intended to convince readers of the truth of the statement of the theorem beyond any doubt, and from which arguments a formal symbolic proof can in principle be constructed. Such arguments are typically easier to check than purely symbolic ones—indeed, many mathematicians would express a preference for a proof that not only demonstrates the validity of a theorem, but also explains in some way why it is obviously true. In some cases, a picture alone may be sufficient to prove a theorem. Because theorems lie at the core of mathematics, they are also central to its aesthetics. Theorems are often described as being "trivial", or "difficult", or "deep", or even "beautiful". These subjective judgments vary not only from person to person, but also with time: for example, as a proof is simplified or better understood, a theorem that was once difficult may become trivial. On the other hand, a deep theorem may be simply stated, but its proof may involve surprising and subtle connections between disparate areas of mathematics. Fermat's Last Theorem is a particularly well-known example of such a theorem.

Read more about Theorem:  Informal Accounts of Theorems, Relation To Proof, Theorems in Logic, Relation With Scientific Theories, Terminology, Layout, Lore, Formalized Account of Theorems

Other articles related to "theorem, theorems":

Spectral Theorem
... linear algebra and functional analysis, the spectral theorem is any of a number of results about linear operators or about matrices ... In broad terms the spectral theorem provides conditions under which an operator or a matrix can be diagonalized (that is, represented as a diagonal matrix in ... In general, the spectral theorem identifies a class of linear operators that can be modelled by multiplication operators, which are as simple as one can hope to find ...
Spectral Theorem - Bounded Self-adjoint Operators
... See also Eigenfunction and Self-adjoint operator#Spectral theorem The next generalization we consider is that of bounded self-adjoint operators on a Hilbert space ... let A be the operator of multiplication by t on L2, that is Theorem ... operator and There is also an analogous spectral theorem for bounded normal operators on Hilbert spaces ...
Robertson–Seymour Theorem
... In graph theory, the Robertson–Seymour theorem (also called the graph minor theorem) states that the undirected graphs, partially ordered by the graph minor relationship, form a well-quasi-ordering ... finite set of forbidden minors, in the same way that Wagner's theorem characterizes the planar graphs as being the graphs that do not have the complete graph K5 and the complete bipartite graph K3,3 ... The Robertson–Seymour theorem is named after mathematicians Neil Robertson and Paul D ...
Robertson–Seymour Theorem - Finite Form of The Graph Minor Theorem
... Friedman, Robertson Seymour (1987) showed that the following theorem exhibits the independence phenomenon by being unprovable in various formal systems that are much stronger than Peano arithmetic, yet being provable ...
Formalized Account of Theorems - Derivation of A Theorem
... The notion of a theorem is very closely connected to its formal proof (also called a "derivation") ... rule) for is Any occurrence of "A" in a theorem may be replaced by an occurrence of the string "AB" and the result is a theorem ... Theorems in are defined as those formulae which have a derivation ending with that formula ...

Famous quotes containing the word theorem:

    To insure the adoration of a theorem for any length of time, faith is not enough, a police force is needed as well.
    Albert Camus (1913–1960)